Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
211 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×S4 |
| commutator, cosocle | C2×C3×SL(2,𝔽3) | C22 |
| Fitting | C2×C3×Q8 | C2×D3 |
| socle | C22×C3 | C2×S4 |
| hypocenter | C3×SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D3×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×C3⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C3×#859 | C2 | semidirect |
| 10. C2×D3×Q8 | D3 | semidirect |
| 17. #859 | D3 | semidirect |
| 24. C3×SL(2,𝔽3) | D4 | semidirect |
| 68. C3×Q8 | #67 | semidirect |
| 82. SL(2,𝔽3) | D12 | semidirect |
| 121. C2×D3 | GL(2,𝔽3) | semidirect |
| 133. Q8 | #626 | semidirect |
| 171. C3 | #5421 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×SL(2,𝔽3) | 2 |
| C2×C3⋊GL(2,𝔽3) | 2 |
| C3×#859 | 2 |
| #5421 | 3 |
| #4806 | 3 |
| C2×#626 | 4 |